Spin Dynamics and Orbital State in LaTiO_3

A neutron scattering study of the Mott-Hubbard insulator LaTiO$_{3}$(T$_{{\rm N}}=132$ K) reveals a spin wave spectrum that is well described by anearest-neighbor superexchange constant $J=15.5$ meV and a smallDzyaloshinskii-Moriya interaction ($D=1.1$ meV). The nearly isotropic spin wavespectrum is surprising in view of the absence of a static Jahn-Tellerdistortion that could quench the orbital angular momentum, and it may indicatestrong orbital fluctuations. A resonant x-ray scattering study has uncovered noevidence of orbital order in LaTiO$_{3}$.

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00Spin Dynamics and Orbital State in LaTiO3B. Keimer1,2, D. Casa2, A. Ivanov3, J.W. Lynn4, M. v. Zimmermann5,
J.P. Hill5, D. Gibbs5, Y. Taguchi6, and Y. Tokura61 Max-Planck-Institut fu¨r Festko¨rperforschung, 70569 Stuttgart, Germany
2 Department of Physics, Princeton University, Princeton, NJ 085443 Institut Laue-Langevin, 156X, 38042 Grenoble Cedex 9, France
4 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 208995 Department of Physics, Brookhaven National Laboratory, Upton, NY 11973
6 Department of Applied Physics, University of Tokyo, Tokyo 113, Japan(February 1, 2008)A neutron scattering study of the Mott-Hubbard insulator LaTiO3 (TN = 132 K) reveals a spin wave
spectrum that is well described by a nearest-neighbor superexchange constant J = 15.5 meV and a
small Dzyaloshinskii-Moriya interaction (D = 1.1 meV). The nearly isotropic spin wave spectrum
is surprising in view of the absence of a static Jahn-Teller distortion that could quench the orbital
angular momentum, and it may indicate strong orbital fluctuations. A resonant x-ray scattering
study has uncovered no evidence of orbital order in LaTiO3.In the layered cuprates exemplified by the series
La2−xSrx CuO4+δ, the transition from a 3d9 antiferro-
magnetic (AF) insulator at x = δ = 0 into an uncon-
ventional metallic and superconducting state with in-
creasing hole concentration (x, δ > 0) has received an
enormous amount of attention. The magnetic spectra of
these materials, revealed by inelastic neutron scattering,
have played a key role in efforts to arrive at a theoreti-
cal explanation of this transition. The pseudocubic per-
ovskite La1−xSrxTiO3+δ undergoes an analogous transi-
tion from a 3d1 AF insulator at x = δ = 0 to a metallic
state with increasing hole concentration [1]. In the ti-
tanates, however, the metallic state shows conventional
Fermi liquid behavior, and no superconductivity is found
[1]. Momentum-resolved probes such as angle-resolved
photoemission spectroscopy and inelastic neutron scat-
tering have thus far not been applied to the titanates, and
the origin of the very different behavior of the metallic
cuprates and titanates is still largely unexplored. Here
we report an inelastic neutron scattering and anomalous
x-ray scattering study of the parent compound of the
titanate series, LaTiO3, that provides insight into the
microscopic interactions underlying this behavior.
Orbital degrees of freedom, quenched in the layeredcuprates by a large Jahn-Teller (JT) distortion of the
CuO6 octahedra, are likely to be a key factor in the phe-
nomenology of the titanates. While the TiO6 octahedra
are tilted in a GdFeO3-type structure, their distortion
is small and essentially undetectable in neutron powder
diffraction experiments on LaTiO3 (Ref. [2]). The crys-
tal field acting on the Ti3+ ion is therefore nearly cu-
bic, and heuristically one expects a quadruply degenerate
single-ion ground state with unquenched orbital angular
momentum opposite to the spin angular momentum due
to the spin-orbit interaction. In other perovskites such
as LaMnO3, such spin-orbital degeneracies are broken
by successive orbital and magnetic ordering transitions[3]. In the orbitally and magnetically ordered state of
LaMnO3, the spin wave spectrum is highly anisotropic
reflecting the different relative orientations of the or-
bitals on nearest-neighbor Mn atoms in different crys-
tallographic directions [4].
The reduced ordered moment (µ0 ∼ 0.45µB, Ref. [5])in the G-Type AF structure of LaTiO3 (inset in Fig. 1)
at first sight appears consistent with a conventional sce-
nario in which the orbital occupancies at every site are
established at some high temperature, and the magnetic
degrees of freedom (coupled spin and orbital angular mo-
menta) order at a lower temperature. Full theoretical
calculations, however, generally predict a ferromagnetic
spin structure for LaTiO3 [6]. As in LaMnO3, the spin
dynamics of LaTiO3 are highly sensitive to the orbital oc-
cupancies and can provide important information in this
regard. We find that the exchange anisotropy is small
and hence inconsistent with the presence of an apprecia-
ble unquenched orbital moment. At the same time, syn-
chrotron x-ray scattering experiments have not revealed
any evidence of reflections showing a resonant enhance-
ment at the Ti K-edge, unlike other perovskites in which
orbital order (OO) is present. These observations, along
with previously puzzling Raman scattering data [7], indi-
cate strong fluctuations in the orbital sector of LaTiO3.
The neutron scattering experiments were conducted onthe BT2 and BT4 triple axis spectrometers at the NIST
research reactor and at the IN8 spectrometer at the In-
stitut Laue-Langevin. For excitation energies up to 25
meV, we used high resolution configurations with verti-
cally focusing pyrolytic graphite (PG) (002) monochro-
mator and PG (002) analyser crystals set for final neu-
tron energies of 14.7 meV or 30.5 meV and horizontally
collimated beams at both NIST and ILL. For excitation
energies of 20 meV and higher, we used a double-focusing
analyser on IN8 with open collimations, and with a Cu
(111) monochromator and a PG (002) analyser set for1

a final neutron energy of 35 meV. PG filters were used
in the scattered beam to reduce higher order contamina-
tion. Data obtained on the different spectrometers and
with different configurations were in good agreement.
The sample was a single crystal of volume 0.2 cm3 andmosaicity 0.5◦ grown by the floating zone technique. It is
semiconducting, and neutron diffraction (Fig. 1) shows a
sharp, second-order Ne´el transition at TN = 132 K, im-
plying a highly homogeneous oxygen content (δ ∼ 0.01 in
LaTiO3+δ) [1]. The diffraction pattern is consistent with
the G-type structure found previously [5], and a small
uncompensated moment (∼ 10−2µB per Ti spin at low
temperatures) appears below TN due to spin canting, as
observed by magnetization measurements. The nuclear
structure of LaTiO3 is orthorhombic (space group Pnma
[2]), but the crystal is fully twinned. Because of the
isotropy of the spin wave dispersions (see below), twin-
ning did not influence the neutron measurements. For
simplicity we express the wave vectors in the pseudocubic
notation with lattice constant a ∼ 3.95A˚. In this nota-
tion, AF Bragg reflections are located at (h/2, k/2, l/2)
with h, k, l odd. Data were taken with the crystal in two
different orientations in which wave vectors of the form
(h, h, l) or (h, k, (h+ k)/2), respectively, were accessible.
Fig. 2 shows inelastic neutron scattering data obtainedin constant-qmode with high resolution near the AF zone
center. The dispersing peak shown disappears above the
Ne´el temperature, thus clearly identifying itself as a spin
wave excitation. In Fig. 3, constant-energy scans ob-
tained in the high-intensity configuration with relaxed
resolution are presented. The profile shapes are strongly
influenced by the spectrometer resolution, and a decon-
volution is required to accurately extract the positions
of the spin wave peaks. For the high-resolution config-
uration with only vertical focusing we used the stan-
dard Cooper-Nathans procedure while a Monte-Carlo
ray-tracing routine was used for the doubly focused ge-
ometry [8]. An analytical approximation to the Ti3+form factor and the standard intensity factors for AF
magnons were incorporated in the programs [9]. The
peak positions thus obtained are shown in Fig. 4 in the
(111) direction.
A very good global fit to all data was obtainedby convoluting the resolution function with a sin-
gle spin wave branch of the generic form hω =
zSJ√
(1 + ǫ)2 − γ2 where hω is the spin wave energy,z = 6 is the coordination number, S = 1/2 is the
Ti spin, J = 15.5 ± 1 meV is the isotropic (Heisen-
berg) part of the nearest-neighbor superexchange [10],
γ = 13
[cos(qxa) + cos(qya) + cos(qza)] with the magnonwave vector q measured from the magnetic zone center,
and the zone center gap is ∆ ∼ zSJ√
2ǫ = 3.3±0.3 meV.The solid lines in Figs. 2-4 result from this global fit and
obviously provide a good description of all data. Inclu-
sion of further-neighbor interactions, damping parame-
ters above the instrumental resolution, or other (nonde-generate) spin wave branches did not improve the fit.
The Heisenberg exchange constant J is in fair agree-ment with predictions based on a comparison of the
Ne´el temperature with numerical simulations (TN =
0.946J/kB ∼ 170 K for spins-1/2 on a simple cubic
lattice [11]). In general, the spin wave gap is deter-
mined by symmetric and antisymmetric (Dzyaloshinskii-
Moriya) anisotropy terms in the superexchange matrix,
by terms originating from direct exchange, by dipolar in-
teractions, and by the single ion anisotropy. The latter
two effects are negligible and nonexistent, respectively, in
spin-1/2 systems. In the GdFeO3 structure, antisymmet-
ric exchange is allowed by symmetry, and its magnitude
is expected to scale with the tilt angle of the TiO6 octahe-
dra. Because of the large tilt angle (11.5◦), we expect this
effect to dominate over the more subtle direct exchange
terms. Theories of superexchange anisotropies [12] were
recently reexamined [13] in the light of neutron scattering
data on the layered cuprates. It was shown that the sym-
metric and antisymmetric terms are related by a hidden
symmetry so that the two zone-center spin wave gaps de-
pend on the relationship between Dzyaloshinkii-Moriya
(DM) vectors centered on each magnetic bond. Specifi-
cally, for two-dimensional (2D) spin structures (such as
the one of La2CuO4) the gaps are degenerate if all DM
vectors have the same magnitude, and the degenerate
gaps are nonzero if, in addition, not all vectors have the
same orientation. The bond-dependent DM vectors for
LaMnO3 (isostructural to LaTiO3) were given in Ref.
[14]. Although a detailed analysis of the spin dynam-
ics for these 3D systems has not been reported, we note
that both of the above criteria are fulfilled for LaTiO3.
In particular, the DM vectors centered on the six Ti-
O-Ti bonds have different orientations but are expected
to have the same magnitude because of the practically
equal bond lengths and bond angles [2]. Our observa-
tion of degenerate but nonzero spin wave gaps therefore
provides support for the predictions of Ref. [13] in a 3D,
non-cuprate system.
Carrying the analogy to the 2D cuprates one step fur-ther, we expect that the anisotropy gap is ∆ ∼ zSD,
and hence D ∼ 1.1 meV is the net DM interaction per Ti
spin. Due to spin canting, the net ferromagnetic moment
per spin should therefore be ∼ µ0D/2J = 1.5× 10−2µB,
in good agreement with the observed value. This sup-
ports our assumption that the DM interaction provides
the dominant contribution to the spin wave gap. A mi-
croscopic calculation of D would be a further interesting
test of the formalism developed in Ref. [13].
The small easy-axis anisotropy in the exchange Hamil-tonian of LaTiO3 is difficult to understand based on sim-
ple crystal-field considerations for the Ti3+ ion. The anti-
symmetric exchange is generally of order (∆g/g)J where
g is the free-electron Lande´ factor and ∆g is its shift in
the crystalline environment [12]. Based on our data, we
therefore estimate ∆g/g ∼ 0.05. On the other hand, in2

the absence of any appreciable static JT distortion as ob-
served experimentally [2], one expects that the spin-orbit
interaction ( Λ ∼ +20 meV [15]) splits the t2g multiplet
of the cubic crystal field Hamiltonian into a quadruply
degenerate ground state and a higher-lying Kramers dou-
blet. In a simple crystal field model, this ground state
is characterized by an unquenched orbital moment equal
and antiparallel to the spin moment (∆g/g = 1). More
elaborate Hartree-Fock calculations [6] do not change this
picture qualitatively: Even if a static JT distortion at the
limits of the experimental error bars [2] is included, the
orbital contribution to the moment remains comparable
to the spin moment (∆g/g ∼ 0.5). There is thus an
order-of-magnitude discrepancy between the predictions
of conventional models and the neutron scattering ober-
vations. The smallness of the spin anisotropy in LaTiO3
is underscored by a different comparison: The D/J ra-
tio of LaTiO3 differs only by a factor of 3 from that
of La2CuO4, whose low-temperature ordered moment is
in near-perfect agreement with the spin-only prediction.
Since D/J scales with the tilt angle of the octahedra
which is a factor of ∼ 3 larger in LaTiO3, the relative
magnitude of this quantity in the two materials can be
accounted for without invoking a large orbital moment
in LaTiO3.
Interestingly, the large discrepancy between the pre-dictions of conventional models and the neutron scat-
tering observations has a close analogy in the electron
spin resonance (ESR) literature. The description of ESR
data on Ti3+ impurities embedded into perovskite lat-
tices in fact commonly requires g-factors that are much
more isotropic than predicted by simple crystal field cal-
culations [15]. According to a widely used model [16],
this is attributed to the dynamical JT effect where the
orbital degeneracy is lifted by coupling to zero-point lat-
tice vibrations.
While the dynamical JT effect is well established inimpurity systems, it has thus far not been reported in
lattice systems which commonly exhibit static, coopera-
tive JT distortions associated with OO. Anomalous x-ray
diffraction with photon energies near an absorption edge
of the transition metal ion has recently been established
as a direct probe of OO in perovskites [3] whose sensitiv-
ity far exceeds conventional diffraction techniques that
probe OO indirectly through associated lattice distor-
tions. We have therefore carried out an extensive search
for reflections characteristic of orbital ordering near the
Ti K-edge (4.966 keV) at beamline X22C at the National
Synchrotron Light Source, with energy resolution ∼ 5
eV. The experiments were performed at low temperature
(T=10K) on a polished (1,1,0) surface of the same crystal
that was also used for the neutron measurements. No ev-
idence for resonant reflections at several high symmetry
positions (such as (12
, 1
2
,0), the ordering wave vector ex-pected for t2g orbitals with a G-type spin structure [17])
was found under the same conditions that enabled theirpositive identification in LaMnO3 [3], YTiO3 [18] and re-
lated materials. If OO is present in LaTiO3, we can hence
conclude that its order parameter is much smaller than in
comparable perovskites. On general grounds, the reduc-
tion of the order parameter should be accompanied by en-
hanced orbital zero-point fluctuations. These may have
already been detected (though not identified as such): A
large electronic background and pronounced Fano-type
phonon anomalies were observed by Raman scattering
in nominally stoichiometric, insulating titanates and are
most pronounced in LaTiO3 [7]. In the light of our obser-
vations, it is of course important to revisit these exper-
iments and rule out any possible role of residual oxygen
defects, inhomogeneity, etc. The presently available data
are, however, naturally interpreted as arising from orbital
fluctuations coupled to lattice vibrations.
An observation not explained by these qualitative con-siderations is the small ordered moment in the AF state.
If the orbital moment is indeed largely quenched, one
would naively expect a spin-only moment of 0.85µB [19],
in contrast to the experimental observation of 0.45µB.
A recently proposed full theory of the interplay between
the orbital and spin dynamics in LaTiO3 has yielded a
prediction of the ordered moment that is in quantitative
agreement with experiment [20].
In conclusion, several lines of evidence from neutron,x-ray and Raman scattering can be self-consistently in-
terpreted in terms of an unusual many body state with
AF long range order but strong orbital fluctuations. This
should be an interesting subject of theoretical research.
The orbital fluctuations are expected to be enhanced in
the presence of itinerant charge carriers and therefore
to strongly influence the character of the insulator-metal
transition. The present study provides a starting point
for further investigations in doped titanates.
We thank A. Aharony, M. Cardona, P. Horsch, D.Khomskii, E. Mu¨ller-Hartmann, A. Oles, G. Sawatzky,
and especially G. Khaliullin for discussions, and J. Kulda
for gracious assistance with his resolution program. The
work was supported by the US-NSF under grant No.
DMR-9701991, by the US-DOE under contrat No. DE-
AC02-98CH10886, and by NEDO and Grants-In-Aid
from the Ministry of Education, Japan.[1] Y. Taguchi et al., Phys. Rev. B 59, 7917 (1999); Y.
Tokura et al., Phys. Rev. Lett. 70, 2126 (1993).[2] D.A. MacLean, H.N. Ng, and J.E. Greedan, J. Solid State
Chem. 30, 35 (1979); M. Eitel and J.E. Greedan, J. Less
Common Met. 116, 95 (1986).[3] Y. Murakami et al., Phys. Rev. Lett. 81, 582 (1998).
[4] F. Moussa et al., Phys. Rev. B 54, 15149 (1996).3

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